THere are many differences, but briefly:
The different castes are: Alphas, Betas, Gammas, Deltas, and Epsilons. THe upper castes are intellegent, and have managerial jobs, where as the lower castes do the manual labor. THe Alpha's have what we would consider the best jobs, and it continues down until the Epsilons, who have the least skilled jobs. The Alphas are tall and fair, while the Epsilons are dark skinned. (because of the prejudices of that time period). They also wear a specific color. For example, the Gammas where green, and the Delta's where Khaki.

One significant difference is that the two highest castes - Beta and Alpha - are not mass-produced, and thus have a degree of individuality. That individuality extends, logically, to physical characteristics as well as to personality. Moreover, just as the highest caste - Alpha - is comprised of the tallest, most beautiful individuals, the Gamma, Delta, and Epsilon workers are progressively shorter and uglier.

Huxley is implying that there is a caste system today, though based on race and economics as opposed to something similar to cloning.

Not only are they different but they are "happy" they belong to their own social class. They have no choice but "be happy" as they have been conditioned to be content with their condition since their birth. One of the key passages in the novel is the description of the sleeping children in their cots, listening to a recording that says that they are frightfully happy not to be alphas (they are too intelligent) or Gammas.

The different castes occur due to their exposure to different chemicals whist being matured in the egg. Alphas and Betas are kept very well, with a lot of oxygen and all good nutrition. On the other hand, Epsilions and Gammas and Deltas are deprived of oxygen to prevent them developing good brains, as well as injected with alcohol. This gives them severe brain handicaps, which the State uses them to keep them happy with their menial jobs, and not striving to further educate themselves.